The Dirichlet problem in Lipschitz domains for higher order elliptic systems with rough coefficients

0.67 MB·English

by  Vladimir Maz'ya

The Dirichlet problem in Lipschitz domains for higher order elliptic
systems with rough coefficients ∗
V. Maz’ya, M. Mitrea and T. Shaposhnikova
7
0
0 Abstract
2
We study the Dirichlet problem in Lipschitz domains and with boundary data in Besov spaces,

n for divergence form strongly elliptic systems of arbitrary order, with bounded, complex-valued
a
coefficients. Ourmainresultgivesasharpconditiononthelocalmeanoscillationsofthecoefficients
J
of the differential operator and the unit normal to the boundary (which is automatically satisfied
1
if these functions belong to space VMO) guaranteeing that the solution operator associated with
3
this problem is an isomorphism.

]
P
A 1 Introduction
.
h
t 1.1 Formulation of the main result
a
m
Afundamentalthemeinthetheory ofpartial differential equations, whichhas profoundandintriguing
[
connections with many other subareas of analysis, is the well-posedness of various classes of boundary

1 value problems under minimal smoothness assumptions on the boundary of the domain and on the
v
coefficients of the corresponding differential operator. The main result of this paper is the solution
8
9 of the Dirichlet problem for higher order, strongly elliptic systems in divergence form, with complex-
8 valued, bounded, measurable coefficients in Lipschitz domains, and for boundary data in Besov spaces,
1
under sharp smoothness assumptions. Inordertobemorespecificweneedtointroducesomenotation.
0
7 Let m,l ∈ N be two fixed integers and, for a bounded Lipschitz domain Ω in Rn, n ≥ 2, with
0 outward unit normal ν = (ν ,...,ν ) consider the Dirichlet problem for the operator
1 n
/
h
at L(X,DX)U := Dα(Aαβ(X)DβU), (1.1)
m
|α|=|β|=m
X
:
v
i.e.,
i
X Dα(A (X)DβU)= 0 for X ∈ Ω,
αβ
r
a  |α|=X|β|=m
(1.2)



 ∂kU
= g on ∂Ω, 0≤ k ≤ m−1.
∂νk k


Here and elsewhere, Dα =(−i∂/∂x1)α1···(−i∂/∂xn)αn if α = (α1,...,αn). The coefficients Aαβ are
l × l matrix-valued functions with measurable, complex-valued entries, for which there exists some
∗2000 Math Subject Classification. Primary: 35G15, 42B20, 35J55, 35J40. Secondary 35J67, 35E05, 46E39.
Keywords: higherorderellipticsystems,Besovspaces,weightedSobolevspaces,meanoscillations,BMO,VMO,Lipschitz
domains, Dirichlet problem, traces, extension operators
TheworkofauthorswassupportedinpartbyfromNSF,DMSandFRGgrantsaswellasfromtheSwedishNational
Science Research Council.
1
finite constant κ > 0 (referred to in the sequel as the ellipticity constant of L) such that
kA k ≤ κ−1 (1.3)
αβ L∞(Ω)
|α|=|β|=m
X
and such that the coercivity condition
ℜ hA (X)DβV(X), DαV(X)idX ≥ κ kDαVk2 (1.4)
αβ L2(Ω)
ZΩ|α|=|β|=m |α|=m
X X
holds for all Cl-valued functions V ∈ C∞(Ω). Throughout the paper, ℜz denotes the real part of
0
z ∈ C and h·,·i stands for the canonical inner product in Cl. Since, generally speaking, ν is merely
bounded and measurable, care should be exercised when defining iterated normal derivatives. For the
setting we have in mind it is natural to take ∂k/∂νk := ( n ξ ∂/∂x )k | or, more precisely,
j=1 j j ξ=ν
∂kU k! P
:= ik ναTr[DαU], 0 ≤ k ≤ m−1, (1.5)
∂νk α!
|α|=k
X
where Tr is the boundary trace operator and να := να1···ναn if α = (α ,...,α ). Now, if p ∈ (1,∞),
1 n 1 n
m,a
a ∈ (−1/p,1 − 1/p) are fixed and ρ(X) := dist(X,∂Ω), a solution for (1.2) is sought in W (Ω),
p
defined as the space of vector-valued functions for which
1/p
|DαU(X)|pρ(X)padX < ∞. (1.6)
(cid:16)|αX|≤mZΩ (cid:17)
In particular, as explained later on, the traces in (1.5) exist in the Besov space Bs(∂Ω), where s :=
p
m,a
1−a−1/p ∈(0,1), for any U ∈ W (Ω). Recall that, with dσ denoting the area element on ∂Ω,
p
|f(X)−f(Y)|p 1/p
f ∈ Bs(∂Ω) ⇔ kfk := kfk + dσ dσ < ∞. (1.7)
p Bps(∂Ω) Lp(∂Ω) |X −Y|n−1+sp X Y
(cid:16)Z∂ΩZ∂Ω (cid:17)
TheabovedefinitiontakesadvantageoftheLipschitzmanifoldstructureof∂Ω. Onsuchmanifolds,
smoothness spaces of index s ∈ (0,1) can be defined in an intrinsic, invariant fashion by lifting their
Euclidean counterparts onto the manifold itself via local charts. Nonetheless, the very nature of the
problem investigated in this paper requires the consideration of higher order smoothness spaces on
∂Ω, in which case the above approach is no longer effective. An alternative point of view has been
introduced by H.Whitney in [63] where he considered higher order Lipschitz spaces on arbitrary
closed sets (see also C.Fefferman’s article [18] for related issues). An extension of this circle of ideas
pertaining to the full scale of Besov and Sobolev spaces on irregular subsets of Rn can be found in the
book [30] by A.Jonsson and H.Wallin. Here we further refine this theory in the context of Lipschitz
domains. For the purpose of this introduction we note that one possible description of these higher
order Besov spaces on the boundary of a Lipschitz domain Ω ⊂ Rn, and for m ∈ N, p ∈ (1,∞),
s ∈ (0,1), reads
B˙m−1+s(∂Ω) = the closure of (i|α|DαV| ) : V ∈C∞(Rn) in Bs(∂Ω) (1.8)
p ∂Ω |α|≤m−1 0 p
n o
(making no notational distinction between a Banach space X and XN = X ⊕ ··· ⊕ X). A formal
definition, which involves higher order Taylor remainders in place of f(X)−f(Y) in (1.7), along with
other equivalent characterizations of B˙m−1+s(∂Ω) can be found in §7.1. Given (1.5)-(1.6) and (1.8), a
p
necessary condition for the boundary data {g } in (1.2) is that
k 0≤k≤m−1
k!
g = ναf , 0 ≤ k ≤ m−1, for some f˙= {f } ∈ B˙m−1+s(∂Ω). (1.9)
k α! α α |α|≤m−1 p
|α|=k
X
2
Let BMO and VMO stand, respectively, for the John-Nirenberg space of functions of bounded
mean oscillations and the Sarason space of functions of vanishing mean oscillations (considered either
on Ω, or on ∂Ω). Our main result, pertaining to the well-posedness of the problem (1.2), then reads
as follows.
Theorem 1.1 Let Ω ⊂ Rn be a bounded Lipschitz domain whose Lipschitz constant is ≤ M, and
assume that the operator L in (1.1) satisfies (1.3)-(1.4). Then there exists c > 0, depending only on
M and the ellipticity constant of L, with the following significance.
Given p ∈ (1,∞) and s ∈ (0,1) set a:= 1−s−1/p. Then the Dirichlet problem (1.2) with boundary
data as in (1.9) has a unique solution U for which (1.6) holds provided the coefficient matrices A
αβ
and the exterior normal vector ν to ∂Ω satisfy
−1
dist(ν,VMO(∂Ω))+ dist(A ,VMO(Ω)) ≤ cs(1−s) p2(p−1)−1+s−1(1−s)−1 . (1.10)
αβ
|α|=X|β|=m (cid:16) (cid:17)
In particular, the above Dirichlet problem is well-posed for each p ∈ (1,∞) and s ∈ (0,1) granted that
A ∈ VMO(Ω) and ν ∈ VMO(∂Ω).
αβ
The above result (along with its inhomogeneous version, presented in §8.1) is sharp. See §8.2 for a
discussion. Note that [ν] + [A ] ≤ cs(1−s)(p2(p−1)−1+s−1(1−s)−1)−1
BMO(∂Ω) |α|=|β|=m αβ BMO(Ω)
suffices for (1.10) to hold. Other corollaries of independent interest are presented below.
P
1.2 Some consequences of the main result and of its proof
In the proof of Theorem 1.1 we shall actually work with a related condition in place of (1.10), which
quantifies the local oscillations of the coefficient matrices and the unit normal. To state this formally,
we need one final piece of terminology. By the BMO mod VMO character of a function F ∈ L (Ω) we
1
shall understand the quantity
{F} := limsup sup − − F(X)−F(Y) dXdY , (1.11)
∗,Ω
ε→0 (Bε)ΩZBε∩Ω ZBε∩Ω(cid:12) (cid:12) !
(cid:12) (cid:12)
(cid:12) (cid:12)
where (B ) stands for the collection of arbitrary balls centered at points of Ω and of radius ε, and
ε Ω
the barred integral is the mean value. In a similar fashion, if (B ) is the collection of n-dimensional
ε ∂Ω
balls with centers on ∂Ω and of radius ε, and if f ∈ L (∂Ω), we set
1
{f} := limsup sup − − f(X)−f(Y) dσ dσ , (1.12)
∗,∂Ω X Y
ε→0 (Bε)∂ΩZBε∩∂Ω ZBε∩∂Ω(cid:12) (cid:12) !
(cid:12) (cid:12)
(cid:12) (cid:12)
For an arbitrary function F we obviously have (with the dependence on the domain dropped) {F} ≤
∗
2dist(F,VMO), where the distance is taken in BMO. Moreover, as a consequence of a result due to
D.Sarason (cf. Lemma 2 on p.393 of [50]), there exists C > 0 such that dist(F,VMO) ≤ C{F} .
∗
Thus, all together, {F} ∼ dist(F,VMO) so that condition (1.10) becomes equivalent to
∗
−1
{ν} + {A } ≤ cs(1−s) p2(p−1)−1+s−1(1−s)−1 . (1.13)
∗,∂Ω αβ ∗,Ω
|α|=X|β|=m (cid:16) (cid:17)
A corollary of our main result is that, under the hypotheses of Theorem 1.1, the problem
Dα(A (X)DβU)= 0 in Ω,
αβ
 |α|=X|β|=m (1.14)


 i|γ|Tr[DγU] = f on ∂Ω, |γ| ≤ m−1,
γ



3
has a unique solution U which satisfies (1.6) whenever
f˙:= {f } ∈ B˙m−1+s(∂Ω). (1.15)
γ |γ|≤m−1 p
However, an advantage of the classical formulation (1.2) over (1.14) is that, in the former case, the
number of boundary conditions is minimal. For a domain Ω ⊂ R2 with boundary of class C1+r,
1 < r < 1, and for real, constant coefficient, scalar operators, the limiting case p = ∞ of the Dirichlet
2
problem (1.14) has been considered by S.Agmon in [2]. Exploiting the special nature of the layer
potentials associated with the equation in the two-dimensional setting, he has proved that there exists
a unique solution U ∈ Cm−1+s(Ω), 0 < s < r, whenever f = i|γ|DγV| , |γ| ≤ m − 1, for some
γ ∂Ω
function V ∈ Cm−1+s(Ω). See also [3] for a related problem.
The innovation that allows us to consider boundary data in Besov spaces as in (1.15) is the
systematic use of weighted Sobolev spaces such as those associated with the norm in (1.6). In relation
to the standard Besov scale in Rn, we would like to point out that, thanks to Theorem 4.1 in [29] on
the one hand, and Theorem 1.4.2.4 and Theorem 1.4.4.4 in [22] on the other, we have
a = 1−s− 1 ∈(0,1−1/p) =⇒ Wm,a(Ω) ֒→ Bm−1+s+1/p(Ω),
p p p
(1.16)
a = 1−s− 1 ∈(−1/p,0) =⇒ Bm−1+s+1/p(Ω) ֒→ Wm,a(Ω).
p p p
Of course, Wm,a(Ω) is just the classical Sobolev space Wm(Ω) when a= 0.
p p
Remarkably, theclassical trace theory for ordinary Sobolev spaces in domains with smooth bound-
ariesturnsouttohaveamostsatisfactory analogueinthisweighted contextandforLipschitzdomains.
Oneof our main results in this regard is identifying the correct class of boundarydata for higher order
m,a
Dirichlet problems for functions in W (Ω). In the process, we establish that
p
U ∈ Wm,a(Ω) and ∂kU = 0 on ∂Ω for 0 ≤ k ≤ m−1
p ∂νk (1.17)
⇐⇒ U belongs to the closure of C∞(Ω) in Wm,a(Ω),
0 p
which provides an answer to the question raised by J.Neˇcas in Problem 4.1 on page 91 of his 1967
book [47]. In the context of unweighted Sobolev spaces and for smoother domains, such a result has
been known for a long time (cf., e.g., P.Grisvard, S.M.Nikol’ski˘ı and H.Triebel’s monographs [22],
[48], [58] and the references therein).
As a consequence of the trace theory developed in §7 we have that, in the context of Theorem 1.1,
1/p
kTr[DαU]k ∼ ρ(X)p(1−s)−1|DαU(X)|pdX , (1.18)
Bs(∂Ω)
p  
|α|≤m−1 |α|≤mZΩ
X X
 
uniformly in U satisfying L(X,D )U = 0 in Ω. The estimate (1.18) can be viewed as a significant
X
generalization of a well-known characterization of the membership of a function to a Besov space
in Rn−1 in terms of weighted Sobolev norm estimates for its harmonic extension to Rn (see, e.g.,
+
Proposition 7′ on p.151 of E.Stein’s book [57]).
1.3 A brief overview of related work
Broadly speaking, there are two types of questions pertaining to the well-posedness of the Dirichlet
problem in a Lipschitz domain Ω for a divergence form, strongly elliptic system (1.1) of order 2m with
boundary data in Besov spaces indexed by s and p.
Question I. Granted that the coefficients of L exhibit a certain amount of smoothness, identify the
indices p, s for which this boundary value problem is well-posed.
4
Question II. Alternatively, having fixed the indices s and p, characterize the smoothness of ∂Ω and of
the coefficients of L for which the aforementioned problem is well-posed.
These, as well as other related issues, have been a driving force behind many exciting, recent devel-
opments in partial differential equations and allied fields. An authoritative account of their impact is
given by C.Kenig in [31] where he describes the state of the art in this field of research up to mid
1990’s. One generic problem which falls under the scope of Question I is to determine the optimal
scale of spaces on which the Dirichlet problem for a strongly elliptic system of order 2m is solvable in
an arbitrary Lipschitz domain Ω in Rn. The most basic case, that of the constant coefficient Laplacian
in arbitrary Lipschitz domains in Rn, is now well-understood thanks to the work of B.Dahlberg and
C.Kenig [14], in the case of L -data, and D.Jerison and C.Kenig [29], in the case of Besov data. The
p
case of (8.15) for boundary data exhibiting higher regularity (i.e., s > 1) has been recently dealt with
by V.Maz’ya and T.Shaposhnikova in [41] where optimal smoothness conditions for ∂Ω are found in
terms of the properties of ν as a Sobolev space multiplier. Generalizations of (8.15) to the case of
variable-coefficient, single, second order elliptic equations have been obtained in [44], [45], [46].
Inspiteofsubstantialprogressinrecentyears,thereremainmanybasicopenquestions,particularly
for l > 1 and/or m > 1, even in the case of constant coefficient operators in Lipschitz domains. In this
context, one significant problem is to determine the sharp range of p’s for which the Dirichlet problem
for strongly elliptic systems with L -boundary data is well-posed. In [49], J.Pipher and G.Verchota
p
have developed a L -theory for real, constant coefficient, higher order systems L = A Dα
p |α|=2m α
when p is near 2, i.e., 2 − ε < p < 2 + ε with ε > 0 depending on the Lipschitz character of Ω.
P
On p.2 of [49] the authors ask whether the L -Dirichlet problem for these operators is solvable in
p
2(n−1)
a given Lipschitz domain for p ∈ (2−ε, +ε), and a positive answer has been recently given
n−3
by Z.Shen in [51]. Let us also mention here the work [1] of V.Adolfsson and J.Pipher who have
dealt with the Dirichlet problem for the biharmonic operator in arbitrary Lipschitz domains and with
data in Besov spaces, [61] where G.Verchota formulates and solves a Neumann-type problem for the
bi-Laplacian in Lipschitz domains and with boundary data in L , [43] where the authors treat the
2
Dirichlet problem for strongly elliptic systems of second order in an arbitrary Lipschitz subdomain
Ω of a (smooth) Riemannian manifold and with boundary data in Bs(∂Ω), when 2−ε < p < 2+ε
p
and 0 < s < 1, as well as the paper [32] by V.Kozlov and V.Maz’ya, which contains an explicit
description of the asymptotic behavior of null-solutions of constant coefficient, higher order, elliptic
operators near points on the boundary of a domain with a sufficiently small Lipschitz constant.
A successful strategy for dealing with Question II consists of formulating and solving the analogue
of the original problem in a standard case, typically when Ω = Rn and L has constant coefficients,
+
and then deviating from this most standard setting by allowing perturbations of a certain magnitude.
A paradigm result in this regard, going back to the work of S.Agmon, A.Douglis, L.Nirenberg and
V.A.Solonnikov inthe50’s and60’s, is thattheDirichlet problemissolvableinthecontext of Sobolev-
Besov spaces if ∂Ω is sufficiently smooth and if L has continuous coefficients. The latter requirement
is an artifact of the method of proof (based on Korn’s trick of freezing the coefficients) which requires
measuring the size of the oscillations of the coefficients in a pointwise sense (as opposed to integral
sense, as in (1.11)). For a version of Question II, corresponding to boundary data selected from
m−1Bm−1−k+s(∂Ω), optimal results have beenobtained by V.Maz’ya andT.Shaposhnikovain [39].
k=0 p
In this context, the natural language for describing the smoothness of the domain Ω is that of Sobolev
Q
space multipliers.
In the smooth context, problems such as (1.2) have been investigated by many authors, including
L.G˚arding [21], M.I.Viˇsik [62], F.E.Browder [7], S.Agmon, A.Douglis and L.Nirenberg [2], [3], [4],
V.Solonnikov [55], [56], L.H¨ormander [27], G.Grubb and N.J.Kokholm [23]. A related result is
as follows. If Ω ⊂ Rn is an arbitrary bounded open set, and g ∈ Wm(Ω), m ∈ N, is given, then the
2
problem
∆mu= 0 in Ω, u∈ Wm(Ω), Dα(u−g)| = 0 for |α| ≤ m−1, (1.19)
2 ∂Ω
5
where the boundary traces are taken in a generalized sense, has a unique solution. Building on some
earlier work of K.Friedrichs, S.L.Sobolev has considered this problem in [53] in the case when ∂Ω
consists of a finite union of submanifolds of Rn of arbitrary codimension. This result also appears
in Sobolev’s 1950 monograph [54]. For arbitrary domains, the well-posedness of (1.19) has been
established by L.I.Hedberg in [24], [25]. The issue of continuity of the variational solution for higher-
order equations at boundary points has been studied by V.Maz’ya in [38].
1.4 Comments on the proof of Theorem 1.1 and the layout of the paper
While the study of boundary value problems for elliptic differential operators with rough coefficients
goes a long way back (it suffices to point to the connections with Hilbert’s 19-th problem and De
Giorgi-Nash-Moser theory), a lot of attention has been devoted lately to the class of operators with
coefficients in VMO. Part of the impetus for the recent surge of interest in this particular line of
work stems from a key observation made by F.Chiarenza, M.Frasca and P.Longo in the early 1990’s.
Morespecifically, whileinvestigating interiorestimates forthesolutionofascalar, second-orderelliptic
differential equation of the form LU = F, these authors have noticed in that U can be related to F
via a potential theoretic representation formula in which the residual terms are commutators between
operators of Mikhlin-Caldero´n-Zygmund type, on the one hand, and operators of multiplication by
the coefficients of L, on the other hand. This made it possible to control these terms by invoking the
commutator estimate of Coifman-Rochberg-Weiss ([12]). An alternative method, based on maximal
operators and good-λ inequalities, has been developed by L.Caffarelli and I.Peral in [10], whereas
whenΩ= Rn, anapproach basedon estimates fortheRiesztransformshasbeendevised byT.Iwaniec
and C.Sbordone in [28]. Further related results can be found in [5], [9], [11], [17], [35].
Compared to the aforementioned works, our approach is more akin to that of F.Chiarenza and
collaborators [11] though there are fundamental differences between solving boundary problems for
higher order and for second order operators. One difficulty inherently linked with the case m > 1
arises from the way the norm in (1.6) behaves under a change of variables κ : Ω = {(X′,X ) : X >
n n
ϕ(X′)} → Rn designedtoflattentheLipschitzsurface∂Ω. Whenm = 1,asimplebi-Lipschitzchanges
+
of variables such as Ω ∋ (X′,X ) 7→ (X′,X − ϕ(X′)) ∈ Rn will do, but matters are considerable
n n +
more subtle in the case m > 1. In this latter situation, we employ a special global flattening map first
introduced by J.Neˇcas (in a different context; cf. p.188 in [47]) and then independently rediscovered
and/or further adapted to new settings by several authors, including V.Maz’ya and T.Shaposhnikova
in [39], B.Dahlberg, C.Kenig J.Pipher, E.Stein and G.Verchota (cf. [13] and thediscussion in [15]),
and S.Hofmann and J.Lewis in [26]. Our main novel contribution in this regard is adapting this
circle of ideas to the context when one seeks pointwise estimates for higher order derivatives of κ and
λ := κ−1 in terms of [∇ϕ]BMO(Rn−1).
Another ingredient of independent interest is deriving estimates for DαDβG(x,y) where G is the
x y
Green function associated with a constant (complex) coefficient system L(D) of order 2m in the upper
half space, which are sufficiently well-suited for deriving commutator estimates in the spirit of [12].
ThemethodsemployedinearlierworksarelargelybasedonexplicitrepresentationformulasforG(x,y)
and,hence, cannotbeadaptedeasily tothecaseof general, non-symmetric, complex coefficient, higher
order systems. By way of contrast, our approach consists of proving directly that the residual part
R(x,y) := G(x,y) − Φ(x− y), where Φ is a fundamental solution for L(D), has the property that
DαDβR(x,y) is a Hardy-type kernel whenever |α| = |β| = m. See also [5] for a discussion of the
x y
difficulties encountered when estimating the residual partR(x,y) in the case when L is not necessarily
symmetric and has complex coefficients.
The layout of the paper is as follows. Section 2 contains estimates for the Green function in the
upper-half space. Section 3 deals with integral operators (of Mikhlin-Caldero´n-Zygmund and Hardy
type) as well as commutator estimates on weighted Lebesguespaces. In thelast partof this section we
also revisit Gagliardo’s extension operator and establish estimates in the context of BMO. Section 4
6
contains a discussion of the Dirichlet problem for higher order, variable coefficient, strongly elliptic
systems in the upper-halfspace. Theadjustments necessary to treat the case of an unboundeddomain
lyingabovethegraphofaLipschitzfunctionarepresentedinSection5,whereasinSection6weexplain
how to handle the case of a bounded Lipschitz domain. In Section 7 we study traces and extension
operators for higher order smoothness spaces on Lipschitz domains. Finally, in Section 8, we deal with
the inhomogeneous version of (1.2); cf. Theorem 8.1 from which Theorem 1.1 follows.
2 Green’s matrix estimates in the half-space
2.1 Statement of the main result
Fix two nonnegative integers m,l and let L(D ) be a matrix-valued differential operator
x
L(D ) = A Dα, (2.1)
x α x
|α|=2m
X
where the A ’s are constant l × l matrices with complex entries. Throughout the paper, Dα :=
α x
i−|α|∂α1∂α2···∂αn if α = (α ,α ,...,α ) ∈ Nn. Here and elsewhere, N stands for the collection of all
x1 x2 xn 1 2 n 0
positive integers and N := N∪{0}. Assume that L is strongly elliptic, i.e., there exists κ > 0 such
0
that kA k ≤ κ−1 and
|α|=m α Cl×l
P ℜhL(ξ)η,ηi ≥ κ|ξ|2mkηk2 , ∀ξ ∈ Rn, ∀η ∈ Cl. (2.2)
Cl Cl
Inwhatfollows, inordertosimplifynotations, weshalldenotethenormsindifferentfinite-dimensional
real Euclidean spaces by |·| irrespective of their dimensions. Also, quite frequently, we shall make
no notational distinction between a space of scalar functions, call it X, and the space of vector-valued
functions (of a fixed, finite dimension) whose components are in X. We denote by F(x) a fundamental
matrix of the operator L(D ), i.e., an l×l matrix solution of the system
x
L(D )F(x) = δ(x)I in Rn, (2.3)
x l
where I is the l×l identity matrix and δ is the Dirac function. We consider the Dirichlet problem
l
L(D )u = f in Rn,
x +
(2.4)
( Tr[∂ju/∂xj]= f j = 0,1,...,m−1, on Rn−1,
n j
where Rn := {x =(x′,x ) : x′ ∈ Rn−1, x > 0} and Tr is the boundary trace operator. Hereafter, we
+ n n
shall identify ∂Rn with Rn−1 in a canonical fashion.
+
For each y′ ∈ Rn−1 we introduce the Poisson matrices P ,...,P for the problem (2.4), i.e., the
0 m−1
solutions of the boundary-value problems
L(D )P (x,y′) = 0I in Rn,
x j l +
(2.5)
 ∂k
 ∂xkPj ((x′,0),y′)= δjkδ(x′ −y′)Il for x′ ∈ Rn−1, 0 ≤ k ≤ m−1,
(cid:18) n (cid:19)
where δ is the usual Kronecker symbol and 0 ≤ j ≤ m−1. The matrix-valued function P (x,0′) is
jk j
positive homogeneous of degree j +1−n, i.e.,
P (x,0′) =|x|j+1−nP (x/|x|,0′), x ∈Rn, (2.6)
j j
where0′ denotestheoriginofRn−1. Therestriction ofP (·,0′)totheupperhalf-sphereSn−1 issmooth
j +
and vanishes on the equator along with all of its derivatives up to order m−1 (see for example, §10.3
in [34]). Hence,
xm
kP (x,0′)k ≤ C n , x∈ Rn, (2.7)
j Cl×l |x|n+m−1−j +
7
and, consequently,
xm
kP (x,y′)k ≤ C n , x∈ Rn, y′ ∈ Rn−1. (2.8)
j Cl×l |x−(y′,0)|n+m−1−j +
By G(x,y) we shall denote the Green’s matrix of the problem (2.4), i.e., the unique solution of the
boundary-value problem
L(D )G(x,y) = δ(x−y)I for x∈ Rn,
x l
 ∂j (2.9)
G ((x′,0),y) = 0I for x′ ∈ Rn−1, 0 ≤ j ≤ m−1,
 ∂xj l
(cid:18) n (cid:19)
where y ∈ Rn is regarded as a parameter. We now introduce the matrix
+
R(x,y) := F(x−y)−G(x,y), x,y ∈ Rn, (2.10)
+
so that, for each fixed y ∈ Rn,
+
L(D )R(x,y) = 0 for x∈ Rn,
x
 ∂j ∂j (2.11)
R ((x′,0),y) = F ((x′,0)−y) for x′ ∈ Rn−1, 0 ≤ j ≤ m−1.
 ∂xj ∂xj
(cid:18) n (cid:19) (cid:18) n (cid:19)
Our goal is to prove the following result.
Theorem 2.1 For all multi-indices α,β of length m
kDαDβR(x,y)k ≤C|x−y¯|−n, (2.12)
x y Cl×l
for x,y ∈ Rn, where y¯:= (y′,−y ) is the reflection of the point y ∈ Rn with respect to ∂Rn.
+ n + +
In the proof of Theorem 2.1 we distinguish two cases, n > 2m and n ≤ 2m, which we shall treat
separately. Our argument pertaining to the situation when n > 2m is based on the following useful
estimate for a parameter dependent integral.
Lemma 2.2 Let a and b be two non-negative numbers and assume that ζ ∈ RN. Then for every ε > 0
and 0< δ < N there exists a constant c(N,ε,δ) > 0 such that
dη c(N,ε,δ)
≤ . (2.13)
RN (|η|+a)N+ε(|η−ζ|+b)N−δ aε(|ζ|+a+b)N−δ
Z
The proof is postponed for §2.4, as to prevent disrupting the flow of the presentation.
2.2 Proof of Theorem 2.1 for n > 2m
In the case when n > 2m there exists a unique fundamental matrix F(x) for the operator (2.1) which
is positive homogeneous of degree 2m−n. We shall use the integral representation formula
R(x,y)= R (x,y)+...+R (x,y), x,y ∈ Rn, (2.14)
0 m−1 +
where R(x,y) has been introduced in (2.10) and, with P as in (2.5), we set
j
∂j
R (x,y) := P (x,ξ′) F ((ξ′,0)−y)dξ′, 0 ≤ j ≤ m−1. (2.15)
j j j
ZRn−1 (cid:18)∂xn (cid:19)
Then, thanks to (2.7) we have
xm dξ′
kR (x,y)k ≤ C n · . (2.16)
j Cl×l Rn−1 |x−(ξ′,0)|n+m−1−j |(ξ′,0)−y|n−2m+j
Z
8
Next, using Lemma 2.2 with
N = n−1 , a = x , δ = 2m−j−1, (2.17)
n
ε = m−j , b = y , ζ = y′−x′,
n
we obtain from (2.16)
j
Cx
n
kR (x,y)k ≤ , 0≤ j ≤ m−1. (2.18)
j Cl×l (|y′−x′|+x +y )n−2m+j
n n
Summing up over j = 0,...,m−1 gives, by virtue of (2.14), the estimate
kR(x,y)k ≤ C|x−y¯|2m−n, x,y ∈ Rn. (2.19)
Cl×l +
Toobtainpointwiseestimates forderivatives of R(x,y), wemake useof thefollowing localestimate
for a solution of problem (2.4) with f = 0. Recall that Ws stands for the classical L -based Sobolev
p p
space of order s. The particle loc is used to brand the local versions of these (and other) spaces.
Lemma 2.3 [see[4]]Letζ andζ befunctionsinC∞(Rn)suchthatζ = 1inaneighborhood ofsuppζ.
0 0 0
Thenthesolutionu∈ Wm(Rn,loc)ofproblem (2.4)withf = 0andf ∈ Wk+1−j−1/p(Rn−1,loc), where
2 + j p
k ≥ m and p ∈ (1,∞), belongs to Wk+1(Rn,loc) and satisfies the estimate
p +
m−1
kζukWpk+1(Rn+) ≤ C kζ0fjkWpk+1−j−1/p(Rn−1)+kζ0ukLp(Rn+) , (2.20)
(cid:16)Xj=0 (cid:17)
where C is independent of u and f .
j
Let B(x,r) denote the ball of radius r > 0 centered at x.
Corollary 2.4 Assume that u ∈ Wm(Rn,loc) is a solution of problem (2.4) with f = 0 and f ∈
2 + j
Ck+1−j(Rn−1,loc). Then for any z ∈ Rn and ρ > 0,
+
m−1k+1−j
sup |∇ u| ≤ C ρ−k sup |u|+ ρs+j−k sup |∇′f | , (2.21)
k s j
Rn+∩B(z,ρ) (cid:16) Rn+∩B(z,2ρ) Xj=0 Xs=0 Rn−1∩B(z,2ρ) (cid:17)
where ∇′ is the gradient of order s in Rn−1. Here C is a constant independent of ρ, z, u and f .
s j
Proof. Given the dilation invariant nature of the estimate we seek, it suffices to assume that ρ = 1.
Given φ ∈ Ck+1−j(Rn−1) supported in Rn−1∩B(z,2), we observe that, for a suitable θ ∈(0,1),
k+1−j
kφk ≤ Ckφkθ kφk1−θ ≤ C sup |∇′φ|. (2.22)
Wpk+1−j−1/p(Rn−1) Lp(Rn−1) Wpk+1−j(Rn−1) s=0 Rn−1∩B(z,2) s
X
Also, if p > n,
sup|∇ v| ≤ Ckvk , (2.23)
Rn k Wpk+1(Rn+)
+
by Sobolev’s inequality. Now, (2.21) follows by combining (2.22), (2.23) with Lemma 2.3. (cid:3)
Given x,y ∈ Rn, setρ:= |x−y¯|/5andpickz ∈ ∂Rn suchthat|x−z| =ρ/2. Itfollows thatforany
+ +
w ∈ Rn ∩B(z,2ρ) we have |x−y¯| ≤ |x−z|+|z−w|+|w−y¯| ≤ ρ/2+2ρ+|w−y¯| ≤ |x−y¯|/2+|w−y¯|.
+
Consequently, |x−y¯|/2 ≤ |w−y¯| for every w ∈ Rn ∩B(z,2ρ), so that, ultimately,
+
C
ρν−k sup k∇′F(w−y)k ≤ , (2.24)
w∈Rn−1∩B(z,2ρ) ν Cl×l |x−y¯|n−2m+k
9
for each ν ∈N . Granted (2.19) and our choice of ρ, we altogether obtain that
0
kDαR(x,y)k ≤ C |x−y¯|2m−n−k, x,y ∈ Rn, ∀α∈ Nn, |α| ≤ k. (2.25)
x Cl×l k + 0
In the following two formulas, it will be convenient to use the notation R for the matrix R associated
L
with the operator L(D ) as in (2.10). By Green’s formula
x
∗
R (y,x) = R (x,y) , x,y ∈ Rn, (2.26)
L L∗ +
h i
where the superscript star indicates adjunction. In order to estimate mixed partial derivatives, we
observe that (2.26) entails
∗
(DβR )(x,y) = (DβR )(y,x) (2.27)
y L x L∗
and remark that L∗ has properties similar to Lh. This, in conicert with (2.25) and the fact that
|x−y¯| = |x¯−y| for x,y ∈ Rn, yields
+
kDβR(x,y)k ≤ C |x−y¯|2m−n−|β|. (2.28)
y Cl×l β
Let us also point out that by formally differentiating (2.11) with respect to y we obtain
L(D )[DβR (x,y)] = 0 for x∈ Rn,
x y L
 ∂j ∂j (2.29)
 DβR ((x′,0),y) = (−D)βF ((x′,0)−y), x′ ∈ Rn−1, 0 ≤ j ≤ m−1.
 j y j
∂x ∂x
(cid:18) n (cid:19) (cid:18) n (cid:19)

With(2.28) and (2.29) in place of (2.19) and (2.11), respectively, we can now run the same program
as above and obtain the estimate
kDαDβR(x,y)k ≤ C |x−y¯|2m−n−|α|−|β|, ∀α,β ∈ N . (2.30)
x y Cl×l αβ 0
2.3 Proof of Theorem 2.1 for n ≤ 2m
When n ≤ 2m we shall use the method of descent. To get started, fix an integer N such that N > 2m
and let (x,z) 7→ G(x,y,z −ζ) denote the Green matrix with singularity at (y,ζ) ∈ Rn×RN−n of the
Dirichlet problem for the operator L(D )+(−∆ )m in the N-dimensional half-space
x z
RN := {(x,z) : z ∈ RN−n, x= (x′,x ), x′ ∈ Rn−1, x > 0}. (2.31)
+ n n
Also, recall that G(x,y) stands for the Green matrix of the problem (2.4).
Lemma 2.5 For all multi-indices α and β of order m and for all x and y in Rn
+
DαDβG(x′,y) = DαDβG(x,y,−ζ)dζ. (2.32)
x y x y
RN−n
Z
Proof. The strategy is to show that
DαDβG(x,y)f (y)dy = DαDβG(x,y,−ζ)dζf (y)dy (2.33)
x y β x y β
Rn Rn RN−n
Z + Z +Z
for each f ∈ C∞(Rn), from which (2.32) clearly follows. To justify (2.33) for a fixed, arbitrary
β 0 +
f ∈ C∞(Rn), we let u be the unique vector-valued function satisfying Dαu ∈ L2(Rn) for all α with
β 0 + +
|α| = m, and such that
L(D )u = Dβf in Rn,
x x β +
 ∂ju (2.34)
(x′,0) = 0 on Rn−1, 0 ≤ j ≤m−1.

 ∂xj
(cid:18) n(cid:19)


10